91Ó°ÊÓ

Many regions along the coast in North and South Carolina and Georgia have experienced rapid population growth over the last 10 years. It is expected that the growth will continue over the next 10 years. This has resulted in many of the large grocery store chains building new stores in the region. The Kelley's Super Grocery Stores, Inc. chain is no exception. The director of planning for Kelley's Super Grocery Stores wants to study adding more stores in this region. He believes there are two main factors that indicate the amount families spend on groceries. The first is their income and the other is the number of people in the family. The director gathered the following sample information. $$\begin{array}{|rrrr|}\hline \text { Family } & \text { Food } & \text { Income } & \text { Size } \\\\\hline 1 & \$5.04 & \$ 73.98 & 4 \\\2 & 4.08 & 54.90 & 2 \\\3 & 5.76 & 94.14 & 4 \\\4 & 3.48 & 52.02 & 1 \\ 5 & 4.20 & 65.70 & 2 \\\6 & 4.80 & 53.64 & 4 \\\7 & 4.32 & 79.74 & 3 \\\8 & 5.04 & 68.58 & 4 \\ 9 & 6.12 & 165.60 & 5 \\\10 & 3.24 & 64.80 & 1 \\ 11 & 4.80 & 138.42 & 3 \\\12 & 3.24 & 125.82 & 1 \\\13 & 6.60 & 77.58 & 7 \\ 14 & 4.92 & 171.36 & 2 \\\15 & 6.60 & 82.08 & 9 \\ 16 & 5.40 & 141.30 & 3 \\\17 & 6.00 & 36.90 & 5 \\\18 & 5.40 & 56.88 & 4 \\\19 & 3.36 & 71.82 & 1 \\\20 & 4.68 & 69.48 & 3 \\\21 & 4.32 & 54.36 & 2 \\\22 & 5.52 & 87.66 & 5 \\\23 & 4.56 & 38.16 & 3 \\ 24 & 5.40 & 43.74 & 7 \\\25 & 4.80 & 48.42 & 5 \\\\\hline\end{array}$$ Food and income are reported in thousands of dollars per year, and the variable "Size" refers to the nümber of people in the household. a. Develop a correlation matrix. Do you see any problems with multicollinearity? b. Determine the regression equation. Discuss the regression equation. How much does an additional family member add to the amount spent on food? c. What is the value of \(R^{2}\) ? Can we conclude that this value is greater than \(0 ?\) d. Would you consider deleting either of the independent variables? e. Plot the residuals in a histogram. Is there any problem with the normality assumption? f. Plot the fitted values against the residuals. Does this plot indicate any problems with homoscedasticity?

Step by step solution

01

Prepare the Data

First, we need to organize the given data on food expenditure, income, and family size. Note that food and income are in thousands of dollars, and size refers to the number of people in a family. This setup will be used to calculate the correlation matrix and build the regression model.

02

Develop the Correlation Matrix

Calculate the correlation coefficients between food expenditure, income, and family size. This is done using the formula for Pearson correlation: \( \text{cor}(X, Y) = \frac{\sum{(X_i - \bar{X})(Y_i - \bar{Y})}}{\sqrt{\sum{(X_i - \bar{X})^2}\sum{(Y_i - \bar{Y})^2}}} \). Compute these values for each pair of variables.

03

Identify Multicollinearity

Examine the correlation matrix to see if the independent variables (income and size) have a high correlation, typically above 0.8 or below -0.8, which would indicate multicollinearity.

04

Develop the Regression Equation

Using linear regression, determine the equation that predicts food expenditure based on income and family size. The equation will be of the form \( \text{Food} = \beta_0 + \beta_1 \cdot \text{Income} + \beta_2 \cdot \text{Size} \). Use a statistical software or calculator to find the coefficients \( \beta_0, \beta_1, \) and \( \beta_2 \).

05

Interpretation of \(\beta_2\)

The slope \( \beta_2 \) represents the change in food expenditure for one additional family member. Discuss this value regarding its implication on food spending.

06

Calculate \(R^2\)

Determine the coefficient of determination, \( R^2 \), which shows the proportion of variance in food expenditure explained by the two independent variables, income and size.

07

Evaluate \(R^2\) Significance

Check if \( R^2 > 0 \). A value greater than 0 indicates that the model explains some variation in the food expenditure, implying it is potentially useful.

08

Consideration for Removing Variables

Based on the coefficient values, significance, and multicollinearity findings, decide whether any of the independent variables (income or size) should be removed from the model.

09

Plot Residual Histogram

Create a histogram of the residual errors from the regression model to assess the normality of the residuals, which is a common assumption in regression analysis.

10

Evaluate Residuals for Normality

Check if the residuals are approximately normally distributed. If not, this might indicate a problem with the normality assumption in the regression model.

11

Plot Fitted Values vs. Residuals

Plot a graph with fitted values (predicted food expenditure) on the x-axis and residuals (actual - predicted) on the y-axis to investigate patterns in residual spread.

12

Check for Homoscedasticity

Assess the fitted vs. residuals plot for an even spread around zero. A fan shape or other patterns might indicate heteroscedasticity, which violates regression assumptions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Correlation Matrix

A correlation matrix is a table used to display the correlation coefficients between several variables at once. It helps to understand the relationship between pairs of variables in a dataset. In this exercise, the correlation matrix is used to analyze the connections between family food expenditure, income, and family size. To construct a correlation matrix, we calculate the Pearson correlation coefficient between each pair of variables. The formula used is: \[ \text{cor}(X, Y) = \frac{\sum{(X_i - \bar{X})(Y_i - \bar{Y})}}{\sqrt{\sum{(X_i - \bar{X})^2}\sum{(Y_i - \bar{Y})^2}}} \]

• Positive Correlation: Indicates that as one variable increases, the other tends to increase as well. A value close to 1 suggests a strong positive relationship.
• Negative Correlation: Suggests that as one variable increases, the other tends to decrease. A value close to -1 indicates a strong negative relationship.
• No Correlation: Values around 0 imply no linear relationship.

In this case, the matrix will help determine if there's a close relation between income, family size, and how much is spent on food.

Multicollinearity

Multicollinearity occurs when two or more independent variables in a regression model are highly correlated. This can make interpreting the regression coefficients difficult because it becomes challenging to isolate the effect of one variable. In the context of regression analysis, multicollinearity can inflate the standard errors of the coefficients, which may lead to unreliable estimates. To spot multicollinearity, we look at the correlation matrix. If two independent variables have a high Pearson correlation coefficient (generally above 0.8 or less than -0.8), it suggests multicollinearity might be an issue. Potential problems caused by multicollinearity include:

• Difficulty identifying the impact of each variable on the dependent variable.
• Increased variance of coefficient estimates, potentially leading to incorrect conclusions.
• Reduced reliability of the model.

When multicollinearity is detected, removing or combining variables, or using techniques like ridge regression, may help resolve these issues.

Coefficient of Determination \(R^2\)

The coefficient of determination, also known as \(R^2\), is an essential metric in regression analysis. It represents the proportion of variance in the dependent variable (food expenditure, in this case) that can be explained by the independent variables (income and family size). A higher \(R^2\) value indicates a better fit for the model. The value of \(R^2\) ranges from 0 to 1:

• \(R^2 = 0\): Indicates that the independent variables do not explain any of the variability in the dependent variable.
• \(R^2 = 1\): Suggests that the independent variables explain all the variability in the dependent variable (a perfect fit).

In practical terms:- An \(R^2\) that is significantly greater than 0 implies that the model is useful for predicting the outcome variable, as it accounts for some of the variability.- However, a high \(R^2\) doesn't guarantee a good model. It's crucial to assess it along with other diagnostic plots and statistical measures to ensure reliability.